Beyond quasinormal modes: a complete mode decomposition of black hole perturbations

Abstract

We show that retarded Green's functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or, Euclidean) mode sum converges. The two regions are separated by a lightcone which scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated to the early time region. We illustrate our results for Pöschl-Teller, BTZ, and Schwarzschild. In the case of Schwarzschild, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $Λ\to 0^+$, and exploit recent exact solutions to the Heun connection problem. In each case we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.

Figures (9)

• Figure 1: Computation of a retarded black hole Green's function $G(t,t',r,r')$ by Fourier transforming the frequency-space expression $\widetilde{G}(\omega,r,r')$ using a contour integral. The analytic structure of $\widetilde{G}(\omega,r,r')$ consists of poles corresponding to QNM modes, and, depending on the black hole, branch cut contributions (not shown here). Closing the contour in the LHP leads to a QNM mode sum which diverges in some spacetime regions. Our work addresses this issue.
• Figure 2: The anatomy of a retarded black hole Green's function. a) The split of the frequency space Green's function $\widetilde{G}(\omega, r, r')$ into two terms $\widetilde{G}_+$ and $\widetilde{G}_-$ based on connection formulae. Both terms contain poles at Matsubara frequencies (red points) with opposite sign for their residues, so that these poles cancel in $\widetilde{G}(\omega, r, r')$. Only $\widetilde{G}_+$ contains QNM poles (black points). b) Computing the Fourier transform of $\widetilde{G}(\omega, r, r')$ in different spacetime regions. Different regions require different contour closure choices so that arc contributions vanish, leading to different convergent residue (mode) sums in each case. The leftmost plot shows a portion of the conformal diagram, with the star corresponding to the delta function location, and the red lines are the lightrays which delineate different regions. In region I) both terms must be closed in the LHP, leading to a cancellation of the MM contributions. Thus $G(t,t',r,r')$ is given by a convergent sum of QNMs only. In region II)$\widetilde{G}_+$ requires UHP closure and $\widetilde{G}_-$ LHP closure. There are no QNM contributions and $G(t,t',r,r')$ is given by a convergent sum of MMs only. In region III) both terms must be closed in the UHP, the residue contributions cancel leading to a zero as required by causality. An analogous construction appears in asymptotically AdS spacetimes, where lightrays reflect off the timelike boundary instead (see FIG.\ref{['fig:btzregions']}).
• Figure 3: The retarded Green's function $|G(t,t',r,r')|$ for small $\Lambda$ Schwarzschild-de Sitter as a convergent mode sum at fixed $r,r'$. There is no fitting, this is directly evaluated by summing residues of $\widetilde{G}_\pm$. In region I (blue) the Green's function is a convergent sum of QNMs. Here we have included the first 6 off-axis QNMs and the first 6 on-axis QNMs in the sum. In region II (green) it is a convergent sum of MMs. Here we have included 30 UHP MMs, 30 LHP MMs and the zero mode residue. In region III (red) it is zero. The lines are drawn as solid in the regions where they apply, and also extended to regions where they do not as dashed lines. For comparison, we show a numerical solution to the Green's function PDE, \ref{['GreensPDE']} (black), where the delta function is approximated by a Gaussian of width $\sigma$ (note that the finite width of the Gaussian gives rise to the non-zero support of the black curve in region III). Parameters used are $s=2$, $l=2$, $R_h = 1$, $\Lambda = 10^{-3}$, $r_* = 12.5$, $r_*' = 10$, and $\sigma = 1/10$.
• Figure 4: The retarded Green's function $G(t,t',z,z')$ for Pöschl-Teller as a convergent mode sum at fixed $z,z'$. In region I (blue) the Green's function is a convergent sum of QNMs, see section \ref{['sec:PTreg1']}. In region II (green) it is a convergent sum of MMs, see section \ref{['sec:PTreg2']}. In region III (red) it is zero, see section \ref{['sec:PTreg3']}. The lines show sums of the first 30 modes in each sum -- in the region where they apply (solid), and their extension to the regions where they do not (dashed). Black points show the Green's function obtained by direct numerical integration for comparison. The parameters chosen are $z' = 8/10$, $z=9/10$, $\nu = i \sqrt{3}/2$.
• Figure 5: Conformal diagram for asymptotically AdS black hole spacetimes showing three regions delineated by lightrays (red lines) emanating from a delta insertion source (red star) and reflecting off the timelike boundary ($\partial$AdS). In regions I and II, the retarded $G$ is constructed from a convergent sum of QNMs and a convergent sum of MMs, respectively. In region III, $G=0$ by causality. For further details, see the analogous construction for asymptotically flat and dS black holes in FIG. \ref{['fig:introimage']}.